Subdifferential Test for Optimality

TL;DR

This paper introduces a new subdifferential-based test for optimality in Banach spaces, offering necessary and sufficient conditions that unify and extend existing optimality criteria.

Contribution

It establishes a first-order optimality condition using subdifferentials and proves that subdifferential operators are monotone absorbing, becoming maximal monotone in convex cases.

Findings

Subdifferential test provides necessary and sufficient optimality conditions.

Subdifferential operators are shown to be monotone absorbing.

In convex cases, subdifferential operators are maximal monotone.

Abstract

We provide a first-order necessary and sufficient condition for optimality of lower semicontinuous functions on Banach spaces using the concept of subdifferential. From the sufficient condition we derive that any subdifferential operator is monotone absorbing, hence maximal monotone when the function is convex.